Search Results

Edit: OK, I think I figured out what she (Singer) is saying and what QY is asking. The restriction of physically real states to invariant subspaces is, as she points out, necessary such that all obs …

This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO. I've been reading Singer's Li …

First, a bit of background. Consider a Hamiltonian $H$ on a finite-dimensional Hilbert space $\mathcal{H}$. Suppose that $H$ is gauge invariant, i.e. $G^{-1}HG = H$ for all $G$ belonging to a unitary …

I'm a Machine Learning researcher who would like to research applications of group theory in ML. There is a term "Partially Observed Groups" in machine learning theory which has been popularized by re …

I'll go against my better judgment and give an answer. I'll leave you with the caveat that my understanding of this area is very superficial, so there might be mistakes in what follows. The first piec …

Physics is usually performed by means of Hamiltonian or Feynman Path Integral approach. Then, for steady state (time independent, when energy is conserved by Noether theorem), the symmetry of the sys …

You can use group representation theory to reduce the dimensions of the problem. And you can explain energy degeneracies. That's all!

I suppose an answer to your question, as simple as possible is this: You would probably be happier if not an irreducible representation, but rather a single function was declared the object of intere …

Many commentators (e.g. Jaynes, Rota) argue that the notion of "differential entropy" is problematic (as commonly defined by $ h(X) = \int ( \log\frac{1}{p(x)} ) p(x) \, dx $, where $X$ is a random va …

I would rather look at this from the opposite end. The key difference between the discrete and continuous cases is that in the discrete case there is a canonical reference measure, namely the counting …

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make. where 'r …

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a way to sample unifor …

Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric $$g'(X,Y)=\int_Gg …

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_1) = 0, E(X_1^2) = \sigma^2 >0, E(|X_1|^3) = \rho < \infty$. Let $Y_n = \frac{1}{n} \sum_{i=1}^n X_i$ and let us note $F_n$ (resp. $\Phi$) t …

I'm wondering what the relation of category theory to programming language theory is. I've been reading some books on category theory and topos theory, but if someone happens to know what the connect …